Fubini's theorem on differentiation

In mathematics, Fubini's theorem on differentiation, named after Guido Fubini, is a result in real analysis concerning the differentiation of series of monotonic functions. It can be proven by using Fatou's lemma and the properties of null sets.[1]

Statement

Assume ${\displaystyle I\subseteq \mathbb {R} }$ is an interval and that for every natural number k, ${\displaystyle f_{k}:I\to \mathbb {R} }$ is an increasing function. If,

${\displaystyle s(x):=\sum _{k=1}^{\infty }f_{k}(x)}$

exists for all ${\displaystyle x\in I,}$ then for almost any ${\displaystyle x\in I,}$ the derivatives exist and are related as:[1]

${\displaystyle s'(x)=\sum _{k=1}^{\infty }f_{k}'(x).}$

In general, if we don't suppose f_k is increasing for every k, in order to get the same conclusion, we need a stricter condition like uniform convergence of ${\displaystyle \sum _{k=1}^{n}f_{k}'(x)}$ on I for every n.[2]

References

1. Jones, Frank (2001), Lebesgue Integration on Euclidean Space, Jones and Bartlett publishers, pp. 527–529.
2. Rudin, Walter (1976), Principles of Mathematical Analysis, McGraw-Hill, p. 152.